L(s) = 1 | + (−0.258 + 0.965i)3-s + (−0.866 − 0.499i)9-s − 1.73i·11-s + (0.707 − 0.707i)17-s − i·19-s + (0.707 − 0.707i)27-s + (1.67 + 0.448i)33-s − 1.73i·41-s + i·49-s + (0.500 + 0.866i)51-s + (0.965 + 0.258i)57-s + (1.22 + 1.22i)67-s + (−1.22 + 1.22i)73-s + (0.500 + 0.866i)81-s + (−0.707 − 0.707i)83-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)3-s + (−0.866 − 0.499i)9-s − 1.73i·11-s + (0.707 − 0.707i)17-s − i·19-s + (0.707 − 0.707i)27-s + (1.67 + 0.448i)33-s − 1.73i·41-s + i·49-s + (0.500 + 0.866i)51-s + (0.965 + 0.258i)57-s + (1.22 + 1.22i)67-s + (−1.22 + 1.22i)73-s + (0.500 + 0.866i)81-s + (−0.707 − 0.707i)83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9837841645\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9837841645\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + 1.73iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.981141431853670685902597457995, −8.662954282247143403699133348525, −7.64265509446526429745478024923, −6.65402131308746304542760961555, −5.73441374658461554786096323773, −5.29485431883283806126914525487, −4.26407550247963211491541406035, −3.38627239398283981752270106147, −2.69703380465988015137939010187, −0.72984464960770401999361856084,
1.43504208371154466812001841293, 2.17493723178632031920049787667, 3.40077123692003507962726773793, 4.52455567240609679922249908986, 5.36000447310506801523251350229, 6.23865926289493646648032473090, 6.87611907826001377666957402361, 7.75757282662236714447159460637, 8.074146104563614812938723738823, 9.190298158538977996901692239652